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Mirrors > Home > MPE Home > Th. List > Mathboxes > pwpwuni | Structured version Visualization version GIF version |
Description: Relationship between power class and union. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
pwpwuni | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwg 4542 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ 𝐴 ⊆ 𝒫 𝐵)) | |
2 | sspwuni 5034 | . . 3 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵) | |
3 | 2 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵)) |
4 | uniexg 7585 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | |
5 | elpwg 4542 | . . . 4 ⊢ (∪ 𝐴 ∈ V → (∪ 𝐴 ∈ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵)) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ∈ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵)) |
7 | 6 | bicomd 222 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ⊆ 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵)) |
8 | 1, 3, 7 | 3bitrd 305 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2110 Vcvv 3431 ⊆ wss 3892 𝒫 cpw 4539 ∪ cuni 4845 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-11 2158 ax-ext 2711 ax-sep 5227 ax-un 7580 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1545 df-ex 1787 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-ral 3071 df-v 3433 df-in 3899 df-ss 3909 df-pw 4541 df-uni 4846 |
This theorem is referenced by: psmeasurelem 43971 |
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